On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Table 2

Bend quintets for Apollonian integer disk packings for principal curvatures 1 through 32.

1 … (0, 0, 1, 1, 1)	1 odd (17, 18, 306, 307, 307) 0	1 odd (26, 27, 702, 703, 703) 0
1 even (1, 2, 2, 3, 3) 0	2 even (17, 19, 162, 162, 166) 1	2 odd (26, 30, 195, 199, 199) 0
1 odd (2, 3, 6, 7, 7) 0	3 skew (17, 22, 75, 78, 82) 2/5	3 skew (26, 31, 162, 163, 171) 4/5
1 odd (3, 4, 12, 13, 13) 0	4 skew (17, 27, 46, 54, 58) 1/5	4 skew (26, 34, 111, 115, 123) 1/2
2 even (3, 5, 8, 8, 12) 1	5 skew (17, 30, 42, 43, 67) 12/13	5 skew (26, 43, 66, 79, 87) 4/17
1 odd (4, 5, 20, 21, 21) 0	1 odd (18, 19, 342, 343, 343) 0	6 skew (26, 51, 55, 66, 94) 14/25
2 even (4, 8, 9, 9, 17) 1	2 odd (18, 22, 99, 103, 103) 0	7 skew (26, 46, 63, 67, 99) 4/5
1 odd (5, 6, 30, 31, 31) 0	3 skew (18, 23, 83, 86, 90) 2/5	1 odd (27, 28, 756, 757, 757) 0
2 even (5, 7, 18, 18, 22) 1	4 skew (18, 31, 43, 54, 58) 2/13	2 even (27, 29, 392, 392, 396) 1
1 odd (6, 7, 42, 43, 43) 0	5 skew (18, 26, 59, 63, 71) 1/2	3 skew (27, 32, 173, 176, 180) 2/5
2 odd (6, 10, 15, 19, 19) 0	6 skew (18, 27, 55, 58, 70) 2/3	4 skew (27, 37, 100, 108, 112) 1/5
3 skew (6, 11, 14, 15, 23) 4/5	7 skew (18, 35, 38, 47, 63) 8/17	5 skew (27, 36, 109, 112, 124) 2/3
1 odd (7, 8, 56, 57, 57) 0	1 odd (19, 20, 380, 381, 381) 0	6 skew (27, 45, 68, 80, 92) 1/3
2 even (7, 9, 32, 32, 36) 1	2 even (19, 21, 200, 200, 204) 1	7 skew (27, 40, 85, 88, 108) 10/13
3 skew (7, 12, 17, 20, 24) 2/5	3 skew (19, 24, 92, 93, 101) 4/5	8 skew (27, 53, 56, 72, 92) 5/13
1 odd (8, 9, 72, 73, 73) 0	4 skew (19, 29, 56, 60, 72) 2/5	9 skew (27, 44, 72, 77, 101) 12/17
2 skew (8, 13, 21, 24, 28) 2/5	5 skew (19, 32, 48, 53, 69) 8/13	10 skew (27, 52, 61, 64, 108) 22/25
3 even (8, 12, 25, 25, 33) 1	6 skew (19, 36, 44, 45, 77) 16/17	1 odd (28, 29, 812, 813, 813) 0
1 odd (9, 10, 90, 91, 91) 0	1 odd (20, 21, 420, 421, 421) 0	2 odd (28, 44, 77, 93, 93) 0
2 even (9, 11, 50, 50, 54) 1	2 odd (20, 36, 45, 61, 61) 0	3 skew (28, 33, 185, 188, 192) 2/5
3 skew (9, 14, 26, 27, 35) 4/5	3 even (20, 24, 121, 121, 129) 1	4 even (28, 32, 225, 225, 233) 1
4 skew (9, 18, 19, 22, 34) 2/3	4 skew (20, 33, 52, 57, 73) 8/13	5 skew (28, 41, 89, 96, 108) 6/13
1 odd (10, 11, 110, 111, 111) 0	5 skew (20, 37, 45, 52, 72) 10/17	6 skew (28, 53, 60, 77, 93) 8/25
2 odd (10, 14, 35, 39, 39) 0	6 even (20, 36, 49, 49, 81) 1	7 skew (28, 48, 69, 77, 101) 3/5
3 skew (10, 18, 23, 27, 35) 1/2	1 odd (21, 22, 462, 463, 463) 0	8 skew (28, 45, 77, 80, 108) 14/17
1 odd (11, 12, 132, 133, 133) 0	2 odd (21, 30, 70, 79, 79) 0	9 even (28, 44, 81, 81, 113) 1
2 even (11, 13, 72, 72, 76) 1	3 even (21, 23, 242, 242, 246) 1	10 skew (28, 57, 60, 65, 113) 24/29
3 skew (11, 16, 36, 37, 45) 4/5	4 skew (21, 34, 55, 66, 70) 2/13	1 odd (29, 30, 870, 871, 871) 0
4 skew (11, 21, 24, 28, 40) 2/3	5 skew (21, 38, 47, 62, 66) 2/17	2 even (29, 31, 450, 450, 454) 1
1 odd (12, 13, 156, 157, 157) 0	6 skew (21, 26, 110, 111, 119) 4/5	3 skew (29, 34, 198, 199, 207) 4/5
2 odd (12, 21, 28, 37, 37) 0	7 skew (21, 30, 71, 74, 86) 2/3	4 skew (29, 42, 94, 103, 111) 4/13
3 skew (12, 17, 41, 44, 48) 2/5	8 skew (21, 31, 66, 70, 82) 3/5	5 skew (29, 39, 114, 118, 130) 2/5
4 even (12, 16, 49, 49, 57) 1	9 skew (21, 39, 46, 58, 70) 1/3	6 skew (29, 46, 79, 90, 102) 6/17
5 skew (12, 21, 29, 32, 44) 2/3	10 even (21, 39, 50, 50, 86) 1	7 skew (29, 54, 63, 82, 94) 6/25
6 skew (12, 25, 25, 28, 48) 10/13	1 odd (22, 23, 506, 507, 507) 0	8 skew (29, 55, 66, 70, 114) 11/13
1 odd (13, 14, 182, 183, 183) 0	2 odd (22, 26, 143, 147, 147) 0	1 odd (30, 31, 930, 931, 931) 0
2 even (13, 15, 98, 98, 102) 1	3 skew (22, 27, 119, 122, 126) 2/5	2 odd (30, 34, 255, 259, 259) 0
3 skew (13, 18, 47, 50, 54) 2/5	4 skew (22, 30, 83, 87, 95) 1/2	3 odd (30, 39, 130, 139, 139) 0
4 skew (13, 23, 30, 38, 42) 1/5	5 skew (22, 39, 51, 62, 74) 6/17	4 odd (30, 55, 66, 91, 91) 0
1 odd (14, 15, 210, 211, 211) 0	6 skew (22, 42, 47, 59, 75) 2/5	5 skew (30, 47, 83, 98, 102) 2/17
2 odd (14, 18, 63, 67, 67) 0	7 skew (22, 35, 62, 63, 87) 12/13	6 skew (30, 38, 143, 147, 155) 1/2
3 skew (14, 19, 54, 55, 63) 4/5	1 odd (23, 24, 552, 553, 553) 0	7 skew (30, 39, 131, 134, 146) 2/3
4 skew (14, 22, 39, 43, 51) 1/2	2 even (23, 25, 288, 288, 292) 1	8 skew (30, 55, 67, 82, 102) 2/5
5 skew (14, 27, 31, 34, 54) 10/13	3 skew (23, 28, 129, 132, 136) 2/5	9 skew (30, 43, 102, 103, 127) 12/13
1 odd (15, 16, 240, 241, 241) 0	4 skew (23, 33, 76, 84, 88) 1/5	10 skew (30, 59, 66, 71, 119) 24/29
2 odd (15, 24, 40, 49, 49) 0	5 skew (23, 36, 64, 73, 81) 4/13	1 odd (31, 32, 992, 993, 993) 0
3 even (15, 17, 128, 128, 132) 1	6 skew (23, 40, 57, 60, 88) 14/17	2 even (31, 33, 512, 512, 516) 1
4 skew (15, 24, 41, 44, 56) 2/3	7 skew (23, 48, 49, 52, 96) 22/25	3 skew (31, 44, 105, 116, 120) 2/13
5 skew (15, 28, 33, 40, 52) 6/13	1 odd (24, 25, 600, 601, 601) 0	4 skew (31, 57, 68, 92, 96) 1/13
6 skew (15, 32, 32, 33, 65) 16/17	2 odd (24, 33, 88, 97, 97) 0	5 skew (31, 36, 224, 225, 233) 4/5
1 odd (16, 17, 272, 273, 273) 0	3 even (24, 28, 169, 169, 177) 1	6 skew (31, 41, 128, 132, 144) 2/5
2 even (16, 20, 81, 81, 89) 1	4 skew (24, 29, 140, 141, 149) 4/5	7 skew, 31, 48, 89, 96) ft= 10/17
3 skew (16, 21, 68, 69, 77) 4/5	5 skew (24, 44, 53, 69, 77) 1/5	8 skew (31, 60, 65, 84, 104) 10/29
4 skew (16, 29, 36, 45, 53) 4/13	6 skew (24, 33, 89, 92, 104) 2/3	9 skew (31, 56, 72, 81, 113) 16/25
5 skew (16, 32, 33, 41, 57) 1/2	7 skew (24, 37, 69, 76, 88) 6/13	1 odd (32, 33, 1056, 1057, 1057) 0
	8 skew (24, 40, 61, 69, 85) 1/2	2 skew (32, 37, 237, 240, 244) 2/5
	9 skew (24, 41, 60, 65, 89) 12/17	3 skew (32, 57, 73, 96, 100) 2/25
	10 skew (24, 49, 49, 60, 88) 14/25	4 even (32, 36, 289, 289, 297) 1
	1 odd (25, 26, 650, 651, 651) 0	5 skew (32, 45, 112, 117, 133) 8/13
	2 even (25, 27, 338, 338, 342) 1	6 skew (32, 48, 97, 105, 121) 1/2
	3 skew (25, 42, 62, 75, 83) 4/17	7 skew (32, 52, 85, 93, 117) 2/3
	4 skew (25, 38, 75, 78, 98) 10/13	8 skew (32, 61, 69, 84, 112) 14/29
	5 skew (25, 50, 51, 66, 86) 2/5	9 skew (32, 49, 96, 97, 129) 16/17
	6 skew (25, 50, 54, 59, 99) 4/5